Physical Realizability Constraints in Spectral Optimization of Passive Analog Networks

We investigate diagonal congruence transformations of symmetric positive-definite conductance matrices in passive analog linear solvers. Under a fixed trace constraint, we distinguish transformations that are mathematically admissible from those realizable using nonnegative resistive conductances and grounding. A grounded star-network family illustrates a strict gap between these optimization domains. For fixed positive grounding conductance h, the ratio of the optimal passive relaxation rate to its unconstrained mathematical counterpart approaches (1 + h)/(2 + h) as the number of outer nodes increases. The result quantifies a structural limitation of passive implementations under an idealized spectral resource budget. We discuss the model’s assumptions and the distinction between spectral convergence and end-to-end computational performance.